Exact Solution for the Conformable Burgers’ Equation by the Hopf-Cole Transform
Yıl 2016,
Cilt: 13 Sayı: 2, - , 01.11.2016
Ali Kurt
Yucel Cenesiz
Orkun Tasbozan
Öz
In this paper, we use Hopf-Cole transform to solve conformable Burgers’ equation. After applying
Hopf-Cole transform to conformable Burgers’ equation, we achieve conformable heat equation. Subsequently
by using Fourier transform we have the exact solution of conformable Burgers’ equation with fractional order
Kaynakça
- [1] H. Bateman, Some Recent Researches on the Motion of Fluids, Monthly Weather Rev. 43, (1915), 163-170.
- [2] L. Debnath, Partial Differential Equations for Scientists and Engineers, Birkh ¨auser, Boston (1997).
- [3] J. M. Burger, Mathematical Examples Illustrating the relations Occurring in the Theory of Turbulent Fluid Motion, Trans. Roy. Neth. Acad. Sci. Amsterdam 17, (1939), 1-53.
- [4] J. M. Burger, A Mathematical Model Illustrating the Theory of Turbulence, in Advances Applied Mechanics 1, Academic Press, New York (1948).
- [5] J. D. Cole, On a Quasi Linear Parabolic Equation Occurring in Aerodynamics, Quart. Apply. Math. 9, (1951), 225-236.
- [6] E. Varoglu, W.D.L. Finn, Space-Time Finite Incorporating Characteristics for the Burgers’ Equation, Internat. J. Number. Methods Engrg. 16, (1980), 171-184.
- [7] D.T. Blackstock, Termaviscous Attenuation of Plane, Peridodic, Finite-amplitude Sound Waves, The Journal of Acoustical Society of America 36, (1964).
- [8] Z.A. Goldberg, Finite-Amplitude Waves in Magnetohydrodynamics, Soviet Physics Jetp 15, (1962), 179-181.
- [9] L.A. Pospelov, Propagation on Finite-amplitude Elastic Waves, Soviet Physics Acoust 11, (1966), 302-304.
- [10] T. Ozis¸, A. Ozdes¸, A Direct Variational Method Applied to Bugers’ Equation, Journal of Computational and Applied Mathematics 71, (1996), 163-175.
- [11] J. Caldwell, P. Wanless, A.E. Cook, A Finite Element Approach to Burgers’ Equation, App. Math. Model. 5, (1981), 189-193.
- [12] D. J. Evans, A.R. Abdullah, The Group Explicit Method for the Solution of the Burgers’ Equation, Computing 32, (1984), 239-253.
- [13] R.C. Mittal, P. Singhal, Numerical Solution of Burger’s Equation, Commun. Numer. Meth. Engng. 9, (1993), 397-406.
- [14] A. Esen, N.M. Yagmurlu, O. Tasbozan, Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations, Appl. Math. Inf. Sci. 7, (2013), 1951-1956.
- [15] E.A.-B. Abdel-Salam, E.A. Yousif, Y.A.S. Arko, E.A.E. Gumma, Solution of Moving Boundary Space-Time Fractional Burger’s Equation, J. App. Math 2014, http://dx.doi.org/10.1155/2014/218092 (2014).
- [16] A. Esen, O. Tasbozan, Numerical Solution of Time Fractional Burgers’ Equation by Cubic B-spline Finite Elements, Mediterr. J. Math., DOI 10.1007/s00009-015-0555-x (2015).
- [17] M. Inc, The Approximate and Exact Solutions of the Space- and Time-fractional Burgers Equations with Initial Conditions by Variational Iteration Method, J. Math. Anal. Appl. 345, (2008), 476-484.
- [18] E. Hopf, The Partial Differential Equation ut +uux =µuxx, Comm. Pure Appl. Math. 3, (1950), 201-230.
- [19] T. Ozis, E.N. Aksan, A. Ozdes, A Finite Element Approach for Solution of Burgers Equation, Applied Mathematics and Computation 139, (2003), 417-428.
- [20] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley &Sons, New York, NY,USA (1993).
- [21] A.Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego (2006).
- [22] I. Podlubny, Fractional Differential Equations. Academic Press,San Diego, Calif, USA (1999).
- [23] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics 264, (2014), 65-70.
- [24] T. Abdeljawad, On Conformable Fractional Calculus, Journal of Computational and Applied Mathematics, 279, (2015), 57-66.
- [25] T. Abdeljawad, Mohammad Al Horani, Roshdi Khalil, Conformable Fractional Semigroup Operators, Journal of Semigroup Theory and Applications 2015 (2015) Article ID. 7.
- [26] D. R. Anderson, D. J. Ulness, Newly Defined Conformable Derivatives, Advances in Dynamical Systems and Applications 10, (2015), 109-137.
Yıl 2016,
Cilt: 13 Sayı: 2, - , 01.11.2016
Ali Kurt
Yucel Cenesiz
Orkun Tasbozan
Kaynakça
- [1] H. Bateman, Some Recent Researches on the Motion of Fluids, Monthly Weather Rev. 43, (1915), 163-170.
- [2] L. Debnath, Partial Differential Equations for Scientists and Engineers, Birkh ¨auser, Boston (1997).
- [3] J. M. Burger, Mathematical Examples Illustrating the relations Occurring in the Theory of Turbulent Fluid Motion, Trans. Roy. Neth. Acad. Sci. Amsterdam 17, (1939), 1-53.
- [4] J. M. Burger, A Mathematical Model Illustrating the Theory of Turbulence, in Advances Applied Mechanics 1, Academic Press, New York (1948).
- [5] J. D. Cole, On a Quasi Linear Parabolic Equation Occurring in Aerodynamics, Quart. Apply. Math. 9, (1951), 225-236.
- [6] E. Varoglu, W.D.L. Finn, Space-Time Finite Incorporating Characteristics for the Burgers’ Equation, Internat. J. Number. Methods Engrg. 16, (1980), 171-184.
- [7] D.T. Blackstock, Termaviscous Attenuation of Plane, Peridodic, Finite-amplitude Sound Waves, The Journal of Acoustical Society of America 36, (1964).
- [8] Z.A. Goldberg, Finite-Amplitude Waves in Magnetohydrodynamics, Soviet Physics Jetp 15, (1962), 179-181.
- [9] L.A. Pospelov, Propagation on Finite-amplitude Elastic Waves, Soviet Physics Acoust 11, (1966), 302-304.
- [10] T. Ozis¸, A. Ozdes¸, A Direct Variational Method Applied to Bugers’ Equation, Journal of Computational and Applied Mathematics 71, (1996), 163-175.
- [11] J. Caldwell, P. Wanless, A.E. Cook, A Finite Element Approach to Burgers’ Equation, App. Math. Model. 5, (1981), 189-193.
- [12] D. J. Evans, A.R. Abdullah, The Group Explicit Method for the Solution of the Burgers’ Equation, Computing 32, (1984), 239-253.
- [13] R.C. Mittal, P. Singhal, Numerical Solution of Burger’s Equation, Commun. Numer. Meth. Engng. 9, (1993), 397-406.
- [14] A. Esen, N.M. Yagmurlu, O. Tasbozan, Approximate Analytical Solution to Time-Fractional Damped Burger and Cahn-Allen Equations, Appl. Math. Inf. Sci. 7, (2013), 1951-1956.
- [15] E.A.-B. Abdel-Salam, E.A. Yousif, Y.A.S. Arko, E.A.E. Gumma, Solution of Moving Boundary Space-Time Fractional Burger’s Equation, J. App. Math 2014, http://dx.doi.org/10.1155/2014/218092 (2014).
- [16] A. Esen, O. Tasbozan, Numerical Solution of Time Fractional Burgers’ Equation by Cubic B-spline Finite Elements, Mediterr. J. Math., DOI 10.1007/s00009-015-0555-x (2015).
- [17] M. Inc, The Approximate and Exact Solutions of the Space- and Time-fractional Burgers Equations with Initial Conditions by Variational Iteration Method, J. Math. Anal. Appl. 345, (2008), 476-484.
- [18] E. Hopf, The Partial Differential Equation ut +uux =µuxx, Comm. Pure Appl. Math. 3, (1950), 201-230.
- [19] T. Ozis, E.N. Aksan, A. Ozdes, A Finite Element Approach for Solution of Burgers Equation, Applied Mathematics and Computation 139, (2003), 417-428.
- [20] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley &Sons, New York, NY,USA (1993).
- [21] A.Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego (2006).
- [22] I. Podlubny, Fractional Differential Equations. Academic Press,San Diego, Calif, USA (1999).
- [23] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics 264, (2014), 65-70.
- [24] T. Abdeljawad, On Conformable Fractional Calculus, Journal of Computational and Applied Mathematics, 279, (2015), 57-66.
- [25] T. Abdeljawad, Mohammad Al Horani, Roshdi Khalil, Conformable Fractional Semigroup Operators, Journal of Semigroup Theory and Applications 2015 (2015) Article ID. 7.
- [26] D. R. Anderson, D. J. Ulness, Newly Defined Conformable Derivatives, Advances in Dynamical Systems and Applications 10, (2015), 109-137.